document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1074904>Question 1074904</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An ellipse and a hyperbola have the same foci, $A$ and $B$, and intersect at four points. The ellipse has major axis 50, and minor axis 40. The hyperbola has conjugate axis of length 20. Let $P$ be a point on both the hyperbola and ellipse. What is $PA \cdot PB$? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689596 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=689596\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If P is a point in an ellipse with foci A and B, and major axis 50,<BR>\n" );
document.write( "by the definition of ellipse,<BR>\n" );
document.write( "PA+PB=50.<BR>\n" );
document.write( "If the minor axis of that ellipse is 40,<BR>\n" );
document.write( "them the semimajor and senior acrs are respectively<BR>\n" );
document.write( "a=25 and b=20.<BR>\n" );
document.write( "That makes the focal distance<BR>\n" );
document.write( "c=15.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For a hyperbola, the focal distance,<BR>\n" );
document.write( "the distance between the vertices (the transverse axis),<BR>\n" );
document.write( "and the conjugate axis are related by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where c is the focal distance,<BR>\n" );
document.write( "2b is the conjugate axis, and<BR>\n" );
document.write( "2a is the transverse axis.<BR>\n" );
document.write( "So for this hyperbola, 2b=20, so b=10,<BR>\n" );
document.write( "and c=15, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=sqrt%2815%5E2-10%5E2%29=sqrt%28225-100%29=sqrt%2825%29=5 ALIGN=MIDDLE  ALT=\"a=sqrt%2815%5E2-10%5E2%29=sqrt%28225-100%29=sqrt%2825%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That makes 2a=10.<BR>\n" );
document.write( "That is the distance between the vertices.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If P is a point in a hyperbola with foci A and B,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28PA-PB%29 ALIGN=MIDDLE  ALT=\"abs%28PA-PB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= distance between the vertices.<BR>\n" );
document.write( "So, for this hyperbola<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28PA-PB%29=10 ALIGN=MIDDLE  ALT=\"abs%28PA-PB%29=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28PA-PB%29%5E2=100 ALIGN=MIDDLE  ALT=\"%28PA-PB%29%5E2=100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "We knew for the ellipse that<BR>\n" );
document.write( "PA+PB=50 ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28PA%2BPB%29%5E2=2500 ALIGN=MIDDLE  ALT=\"%28PA%2BPB%29%5E2=2500\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4PA%2APB=%28PA%2BPB%29%5E2-%28PA-PB%29%5E2 ALIGN=MIDDLE  ALT=\"4PA%2APB=%28PA%2BPB%29%5E2-%28PA-PB%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4PA%2APB=2500-100=2400 ALIGN=MIDDLE  ALT=\"4PA%2APB=2500-100=2400\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PA%2APB=2400%2F4 ALIGN=MIDDLE  ALT=\"PA%2APB=2400%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28PA%2APB=600%29 ALIGN=MIDDLE  ALT=\"highlight%28PA%2APB=600%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );